Toy models for evolution in many environments and high dimensions

The classic question of statistical physics – what is different when some parameter is large? - acquires a whole new life in the study of evolution. Most of our intuition about evolution and ecology comes from analysis of low-dimensional models, with few environments or few factors determining fitness. What novel phases might arise when evolution is examined in the realistic regime of many environments as opposed to a few? I will describe one minimally structured toy model for these intriguing general questions, and will discuss a few avenues we are investigating in my group. In particular, I will argue that, generically, in high dimensions one no longer expects a direct exposure to some environment to be the most effective way of achieving highest fitness in it.